A296095 Integers represented by cyclotomic binary forms.

3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 25, 26, 27, 28, 29, 31, 32, 34, 36, 37, 39, 40, 41, 43, 45, 48, 49, 50, 52, 53, 55, 57, 58, 61, 63, 64, 65, 67, 68, 72, 73, 74, 75, 76, 79, 80, 81, 82, 84, 85, 89, 90, 91, 93, 97, 98, 100, 101, 103, 104, 106, 108, 109, 111, 112, 113, 116, 117, 121, 122

Offset: 1

Author

Michel Waldschmidt, Feb 14 2018

Keywords

• nonn

Comments

Possibly a subsequence of A000401. - C. S. Davis, May 10 2025

All terms divisible by 11 appear to be either of the form 11^2*A383784(n) for n>1 or x^4 + u*x^3*y + x^2*y^2 + u*x*y^3 + y^4 for x>y>0 and u={-1, 1}. - C. S. Davis, May 14 2025

Links

• Peter Luschny, Table of n, a(n) for n = 1..1000 (terms 1..519 from Michel Waldschmidt).
• Étienne Fouvry, Claude Levesque, and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

Crossrefs

Complement of A293654.

Supersequence of A383784(n) for n>3, according to Proposition 6.2 of Fouvry et al.

Programs

Julia

using Nemo
function isA296095(n)
    n < 3 && return false
    R, z = PolynomialRing(ZZ, "z")
    N = QQ(n)
    # Bounds from Fouvry, Levesque and Waldschmidt
    logn = log(n)^1.161
    K = Int(floor(5.383*logn))
    M = Int(floor(2*(n/3)^(1/2)))
    k = 3
    while true
        c = cyclotomic(k, z)
        e = Int(eulerphi(ZZ(k)))
        if k == 7
            K = Int(ceil(4.864*logn))
            M = Int(ceil(2*(n/11)^(1/4)))
        end
        for y in 2:M, x in 1:y
            N == y^e*subst(c, QQ(x,y)) && return true
        end
        k += 1
        k > K && break
    end
    return false
end
A296095list(upto) = [n for n in 1:upto if isA296095(n)]
println(A296095list(2040)) # Peter Luschny, Feb 28 2018

Maple

with(numtheory): for n from 3 to 1000 do F[n] := expand(y^phi(n)*cyclotomic(n, x/y)) od: for m to 1000 do for n from 3 to 50 do for x from -50 to 50 do for y from -50 to 50 do if `and`(F[n] = m, max(abs(x), abs(y)) > 1) then print(m); m := m+1; n := 3; x := -50; y := -50 end if end do end do end do end do;

Mathematica

isA296095[n_]:=
If[n<3, Return[False],
logn = Log[n]^1.161;
K = Floor[5.383*logn];
M = Floor[2*(n/3)^(1/2)];
k = 3;
While[True,
   If[k==7,
      K = Ceiling[4.864*logn];
      M = Ceiling[2*(n/11)^(1/4)]
   ];
   For[y=2, y<=M, y++,
      p[z_] = y^EulerPhi[k]*Cyclotomic[k,z];
      For[x=1, x<=y, x++, If[n==p[x/y], Return[True]]]
   ];
   k++;
   If[k>K, Break[]]
];
Return[False]
];
Select[Range[122], isA296095] (* Jean-François Alcover, Feb 20 2018, translated from Peter Luschny's Sage script, updated Mar 01 2018 *)

Sage

def isA296095(n):
    if n < 3: return False
    logn = log(n)^1.161
    K = floor(5.383*logn)
    M = floor(2*(n/3)^(1/2))
    k = 3
    while True:
        if k == 7:
            K = ceil(4.864*logn)
            M = ceil(2*(n/11)^(1/4))
        for y in (2..M):
            p = y^euler_phi(k)*cyclotomic_polynomial(k)
            for x in (1..y):
                if n == p(x/y): return True
        k += 1
        if k > K: break
    return False
def A296095list(upto):
    return [n for n in (1..upto) if isA296095(n)]
print(A296095list(122)) # Peter Luschny, Feb 28 2018